# The Envelope Theorem 1

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## The Envelope Theorem

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ya2

### (With the qualification that I have to check that indeed a > 0 at any proposed solution.) In terms of notation, here a is playing the role of “x” and y is playing the role of “q”.

2Formally, define r : U → RN +Lby r(q) = (φ(q), q). Then fv(q) = f (r(q)). By the Chain Rule, Dfv(q) = Df (x, q)Dr(q)

=

Dxf (x, q) Dqf (x, q) 

 Dφ(q) I



= Dxf (x, q)Dφ(q) + Dqf (x, q).

as claimed. Many of the other Chain Rule applications are similar.

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